Solved Examples on Integration
Example 1: Solve: ∫x6dx
Solution:
∫xn dx = xn+1/ (n + 1) + C
∫x6 dx = x6+1/ (6 + 1) + C
∫x6 dx = [x7/ 7] + C
Example 2: Solve: ∫3x+2dx
Solution:
∫3x+2dx = ∫3x+2dx
∫3x+2dx = ∫3x.32dx
∫3x+2dx = 9 ∫3xdx
∫3x+2dx = 9[3x / loge3] + C
Example 3: Solve: ∫(x3 + 4x2 + 3x + 1)dx
Solution:
∫(x3 + 4x2 + 3x + 1)dx = ∫x3 dx + ∫(4x2) dx + ∫(3x) dx + ∫(1) dx
∫(x3 + 4x2 + 3x + 1)dx = [x3+1 / (3 + 1)] + 4 [x2+1 / (2 + 1)] + 3[x1+1 / (1 + 1)] + x + C
∫(x3 + 4x2 + 3x + 1)dx = [x4 / 4] + 4 [x3 / 3] + 3[x2 / 2] + x + C
Example 4: Solve: ∫(x + 3sinx)dx
Solution:
∫(x + 3sinx) dx = ∫x dx + ∫ (3sinx) dx
∫(x + 3sinx) dx = [x1+1 / (1 +1)] + 3(-cos x) + C
∫(x + 3sinx) dx = (x2 / 2) – 3 cos x + C
Example 5: Solve: ∫[1/ (x2 + 25)] dx
Solution:
∫[1/ (x2 + 25)] dx = ∫[1/ (x2 + 52)] dx
∫[1/ (x2 + 25)] dx = (1 / 5) tan-1 (x / 5) + C
Example 6: Solve: ∫(2ex + x3) dx
Solution:
∫(2ex + x3) dx = ∫2ex dx + ∫x3 dx
∫(2ex + x3) dx = 2ex + [x3+1 /(3 +1)] + C
∫(2ex + x3) dx = 2ex + [x4 /4] + C
Integration
Integration is an important part of calculus. It involves finding the anti-derivative of a function and is used to solve integrals. Integration has numerous applications in various fields, such as mathematics, physics, and engineering.
This article serves as a comprehensive guide to integration, covering everything from integration formulas to methods for finding integrals. It also explains the properties and real-world applications of integration through solved examples. Let’s start exploring the topic of Integration.
Table of Content
- What is Integration?
- Integration Definition
- Integration Symbol
- Rules for Integration
- Power Rule of Integration
- Addition Rule of Integration
- Subtraction Rule of Integration
- Constant Multiple Rule of Integration
- Antiderivative: Integration as Inverse Process of Differentiation
- Integration Formulas
- Types of Integration
- Definite Integration
- Indefinite Integration
- Improper Integration
- Integration Techniques
- Integration of Basic Functions
- Integration of Constant Function
- Integration of Trigonometric Functions
- Integration of Exponential and Logarithmic Functions
- Applications of Integration
- Integration in Physics and Engineering
- Integration in Economics and Finance
- Integration vs Differentiation
- Solved Examples on Integration
- Practice Questions on Integration
Contact Us